Marriage: Martha Tate
I resisted his identification as the Robert Smith who married Martha Tate in Wilson County, Tennessee, several counties east of Henry County until the social security record of their son Jacob "Jake" Tate Smith provided persuasive evidence.
Martha possibly died not long after 1880, and Robert may have been married once or twice after her death. Marriages are recorded for Robert H. Smith and Sarah E. Thompson and for Robert Smith and Annie Collins. Robert H.'s marriage to Annie Collins is unlikely since she was under the age of 18 while Robert H would have been nearly 50 in 1898 when the marriage occurred.
Since Robert's family was next door to Catherine and her family in 1880, I believed it somewhat likely that her orphaned children joined his and his wife's family; however, I cannot find more on anyone in the family except Jake after the 1880 census.
Jake's information is here chiefly because his social security data provide the names of his mother and father, which helps explain the family history.
Jake married Allie Jenkins in 1897, and she died between 1905 and 1910 after giving birth to four children. Jake seems never to have married again. The children were James L., born 1898 or 1899, Lula Mae, born 1901 or 1902, Ora, born 1903 or 1904, and Didis, born 1905 or 1906. Lula married and had some children but died in her twenties. Ora married twice but had no children with either husband. Didis marrried but died still in her teens. I found no information about James after the 1910 census.
Robert H. Smith and Sarah E. Thompson, 5 June 1887, Stoddard County, Missouri
Citation: Stoddard County Marriage Records, Volume 2, 1881-1888, page 569 FamilySearch database online, URL: https://www.familysearch.org/ark:/61903/3:1:3QS7-L98M-B34W?i=507&cc=2060668&personaUrl=%2Fark%3A%2F61903%2F1%3A1%3AWP98-7V6Z
Robert Smith and Annie Collins, 6 March 1898, Carter County, Missouri
Citation: Carter County Marriage Records, Volume B, 1890-1898, page 288, FamilySearch database online, URL: https://www.familysearch.org/ark:/61903/3:1:3QSQ-G9CT-LV7C?i=357&cc=2060668&personaUrl=%2Fark%3A%2F61903%2F1%3A1%3AWGTB-W4ZM
Jacob Smith and Allie Jenkins, 6 February 1897, Sumner County, Tennessee
Sumner County Records, County Court Clerk's Office Marriage Records, 1895 - 1902, page 124
1870 Census Wilson Co, TN, p 561B: Robert Smith, 21 TN, Martha Smith, 18 TN
1880 Census Ripley Co page 526C: family of R. H. Smith, 30, TN TN NC, Martha, 27, TN TN TN, J. E., male 9 TN TN TN, M. E., female 6, MO TN TN, J. T., male 2 MO TN TN
Jacob Smith, 32 wid, MO TN TN, James L., 11, TN MO TN, Lula M., 8, TN MO TN, Ora, 6, TN MO TN, Didis, dau, 4, TN MO TN
From US National Archives at https://aad.archives.gov; URL https://www.familysearch.org/ark:/61903/1:1:6KQW-CTKQ
Jake Tate Smith, born 15 October 1877, Doniphan, MO (spelled Donithan on record); parents Robert Smith and Martha E. Tate; issued June 1937